Chapter 34B - Reflection Chapter 34B - Reflection and Mirrors II and Mirrors II (Analytical)(Analytical)

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity© 2007

Objectives: Objectives: After completing After completing this module, you should be this module, you should be

able to:able to:• Define and illustrate the following Define and illustrate the following

terms: terms: realreal and and virtualvirtual images, images, convergingconverging and and divergingdiverging mirrors, mirrors, focal focal lengthlength, and , and magnificationmagnification..

• Predict mathematically the Predict mathematically the naturenature, , sizesize, , and and locationlocation of images formed by of images formed by spherical mirrors.spherical mirrors.

• Understand and apply the Understand and apply the sign sign conventionsconventions that apply to focal lengths, that apply to focal lengths, image distances, image heights, and image distances, image heights, and magnification.magnification.

• Determine mathematically the Determine mathematically the magnificationmagnification and/or the focal length of and/or the focal length of spherical mirrors.spherical mirrors.

Analytical OpticsAnalytical Optics

In this unit, we will discuss analytical In this unit, we will discuss analytical relationships to describe mirror images more relationships to describe mirror images more accurately. But first we will review some accurately. But first we will review some graphical principles covered in Module 34a graphical principles covered in Module 34a on light reflection.on light reflection.

In this unit, we will discuss analytical In this unit, we will discuss analytical relationships to describe mirror images more relationships to describe mirror images more accurately. But first we will review some accurately. But first we will review some graphical principles covered in Module 34a graphical principles covered in Module 34a on light reflection.on light reflection.

The Plane MirrorThe Plane Mirror

Object distanc

e

Image distanc

e

=

p = q

ObjecObjectt

ImageImage

pp qq

Object distance:Object distance: The straight-line The straight-line distance distance pp from the surface of a mirror from the surface of a mirror to the object. to the object. Image distance:Image distance: The straight-line The straight-line distance distance qq from the surface of a mirror from the surface of a mirror to the image. to the image.

Image is virtual

Spherical MirrorsSpherical MirrorsA A spherical mirrorspherical mirror is formed by the is formed by the inside (inside (concaveconcave) ) or outside or outside ((convexconvex) surfaces ) surfaces of a sphere.of a sphere.A A concave concave spherical mirrorspherical mirror is is shown here with shown here with parts identified.parts identified.The The axisaxis and and linear linear apertureaperture are are shown.shown.

Concave Mirror

Radius of Curvature RVertex V

Center of Curvature C

Linear aperture

V

C

R Axis

The Focal Length The Focal Length ff of a of a MirrorMirror

axis

Incident parallel ray

f

The focal length, f

The focal length f is equal to half the radius RThe focal length f is equal to half the radius R

Since Since ii = = rr, we , we find that find that FF is is mid- way mid- way between between VV and and CC; we find:; we find:The focal The focal length length f f is: is:

2

Rf

C Vr

i

RF

Focal point

Converging and Diverging Converging and Diverging MirrorsMirrorsConcave Concave mirrors and mirrors and converging parallel converging parallel rays will be called rays will be called converging mirrorsconverging mirrors..

ConvexConvex mirrors and mirrors and diverging parallel diverging parallel rays will be called rays will be called diverging mirrorsdiverging mirrors..

CF

Converging Mirror

Concave

C F

Diverging Mirror

Convex

DefinitionsDefinitions

Focal length:Focal length: The straight-line distance The straight-line distance ff from the surface of a mirror to focus of from the surface of a mirror to focus of the mirror. the mirror. Magnification:Magnification: The ratio of the size of The ratio of the size of the image to the size of the object.the image to the size of the object.

Real image:Real image: An image formed by real An image formed by real light rays that can be projected on a light rays that can be projected on a screen. screen. Virtual image:Virtual image: An image that appears to An image that appears to be at a location where no light rays be at a location where no light rays reach. reach. Converging and diverging mirrors:Converging and diverging mirrors: Refer to Refer to the reflection of parallel rays from surface the reflection of parallel rays from surface of mirror.of mirror.

Image Construction Image Construction Summary:Summary:

Ray 1:Ray 1: A ray parallel to mirror axis A ray parallel to mirror axis passes through the focal point of a passes through the focal point of a concave mirror or appears to come concave mirror or appears to come from the focal point of a convex mirror.from the focal point of a convex mirror.

Ray 2:Ray 2: A ray passing through the focus A ray passing through the focus of a concave mirror or proceeding of a concave mirror or proceeding toward the focus of a convex mirror is toward the focus of a convex mirror is reflected parallel to the mirror axis. reflected parallel to the mirror axis.

Ray 3:Ray 3: A ray that proceeds along a A ray that proceeds along a radius is always reflected back along its radius is always reflected back along its original path. original path.

C F

Converging mirror

Examples of Image Examples of Image ConstructionConstructionThe three principal rays for both The three principal rays for both converging (concave) and diverging converging (concave) and diverging (convex) mirrors.(convex) mirrors.

Ray 1

Ray 2

Ray 3

CC

Diverging mirror

F

Ray 1

Ray 2

Ray 3

Image

Review of Imaging FactsReview of Imaging FactsFor plane mirrors, the object distance For plane mirrors, the object distance equals the image distance and all images equals the image distance and all images are erect and virtual.are erect and virtual.

For converging mirrors and diverging For converging mirrors and diverging mirrors, the focal length is equal to one-mirrors, the focal length is equal to one-half the radius.half the radius.All images formed from convex mirrors All images formed from convex mirrors are erect, virtual, and diminished in size.are erect, virtual, and diminished in size.

Except for objects located inside the focus Except for objects located inside the focus (which are erect and virtual), all images (which are erect and virtual), all images formed by converging mirrors are real and formed by converging mirrors are real and inverted. inverted.

Questions About ImagesQuestions About Images

3. Is it enlarged, diminished, or the same size?

2. Is the image real or virtual?

1. Is the image erect or inverted?

4. What are object and image distances p and q?

5. What is the height y’ or size of image?6. What is the magnification M = y’/y of image?

Definition of SymbolsDefinition of SymbolsBy applying algebra and geometry to ray-By applying algebra and geometry to ray-tracing diagrams, such as the one below, one tracing diagrams, such as the one below, one can derive a relationship for predicting the can derive a relationship for predicting the location of images.location of images.

y

Y’

R

q

p

f

Object dist. pImage dist. q

Focal length fRadius R

Object size y

Image size y’

2

Rf

Mirror EquationMirror Equation

y

Y’

R

q

p

f

2

Rf

2

Rf

1 1 1

p q f

1 1 1

p q f

The following equations are given The following equations are given without derivation. They apply equally without derivation. They apply equally well for both converging and diverging well for both converging and diverging mirrors.mirrors.

The following equations are given The following equations are given without derivation. They apply equally without derivation. They apply equally well for both converging and diverging well for both converging and diverging mirrors.mirrors.

Sign ConventionSign Convention

1. Object distance p is positive for real objects and negative for virtual objects.

2. Image distance q is positive for real images and negative for virtual images.3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.

1 1 1

p q f

1 1 1

p q f

Example 1.Example 1. A A 6 cm6 cm pencil is placed pencil is placed 50 50 cmcm from the vertex of a from the vertex of a 40-cm40-cm diameter diameter mirror. What are the location and nature mirror. What are the location and nature of the image?of the image?

Sketch the rough Sketch the rough image.image.p p = 50 cm; = 50 cm; RR = 40 = 40 cmcm

40 cm; 20 cm

2 2

Rf f

1 1 1

p q f 1 1 1

50 cm 20 cmq

C F

p

q

f

Example 1 (Cont.).Example 1 (Cont.). What are the location What are the location and nature of the image? (and nature of the image? (p p = 50 cm; = 50 cm; f f = = 2020 cm)cm)

1 1 1

50 cm 20 cmq

1 1 1

20 cm 50 cmq

q = +33.3 cmq = +33.3 cm

The image is real (+q), inverted, diminished, and located 33.3 cm from mirror (between F and C).

The image is real (+q), inverted, diminished, and located 33.3 cm from mirror (between F and C).

C F

p

q

f

Working With Reciprocals:Working With Reciprocals:The mirror equation can The mirror equation can easily be solved by using the easily be solved by using the reciprocal button (reciprocal button (1/x1/x) on ) on most calculators:most calculators:

1 1 1

p q f

1 1 1

p q f

P qP q 1/x1/x ++ 1/x1/x == 1/x1/xFinding Finding f:f:

Same with reverse notation calculators Same with reverse notation calculators might be:might be:

Finding Finding f:f: P qP q 1/x1/x ++1/x1/x 1/x1/xEnteEnterr

Possible sequence for finding Possible sequence for finding ff on linear on linear calculators:calculators:

Be careful with substitution of signed Be careful with substitution of signed numbers!numbers!Be careful with substitution of signed Be careful with substitution of signed numbers!numbers!

Alternative SolutionsAlternative SolutionsIt might be useful to solve the mirror It might be useful to solve the mirror equation algebraically for each of the equation algebraically for each of the parameters:parameters:

1 1 1

p q f

1 1 1

p q f

qpf

q p

qpf

q p

qf

pq f

qfp

q f

pf

qp f

pfq

p f

Example 2:Example 2: An arrow is placed An arrow is placed 30 cm30 cm from the surface of a polished sphere of from the surface of a polished sphere of radius radius 80 cm80 cm. What is the location and . What is the location and nature of image?nature of image?

Draw image Draw image sketch:sketch:p p = 30 cm; = 30 cm; RR = -80 = -80 cmcm

-80 cm; 40 cm

2 2

Rf f

Solve the mirror Solve the mirror equation for equation for qq, then , then watch signs carefully watch signs carefully on substitution:on substitution:

pfq

p f

pfq

p f

Example 2 (Cont.)Example 2 (Cont.) Find location and Find location and nature of image when nature of image when pp = 30 cm and = 30 cm and qq = -40 cm.= -40 cm.

(30 cm)(-40 cm)

30 cm - (-40 cm)q

q = -17.1 cm

The image is The image is virtual virtual (-q), (-q), erecterect, and , and diminisheddiminished. It appears to be located at . It appears to be located at a distance of a distance of 17.1 cm17.1 cm behindbehind the the mirror.mirror.

The image is The image is virtual virtual (-q), (-q), erecterect, and , and diminisheddiminished. It appears to be located at . It appears to be located at a distance of a distance of 17.1 cm17.1 cm behindbehind the the mirror.mirror.

Magnification of ImagesMagnification of ImagesThe The magnificationmagnification MM of an image is the of an image is the ratio of the ratio of the image sizeimage size y’y’ to the to the object object sizesize yy..

Magnification:

'y qM

y p

yy and and y’ y’ are positive when erect; negative are positive when erect; negative inverted.inverted.

Obj.

Img.

Obj.

Img.

M = +2 M = -1/2

y y’

y y’

qq is positive when real; negative when virtual. is positive when real; negative when virtual.

MM is positive when image erect; negative is positive when image erect; negative inverted.inverted.

Example 3.Example 3. An An 8-cm8-cm wrench is placed wrench is placed 10 cm10 cm from a diverging mirror of from a diverging mirror of f = -f = -20 cm20 cm. What is the location and size of . What is the location and size of the image? the image?

( 6.67 cm)

10 cm

qM

p

Magnification:M = +0.667

Since M = y’/y y’ = My or:

(10cm)(-20cm)

10 cm - (-20 cm)

pfq

p f

q = - 6.67 cmq = - 6.67 cm

Virtual Virtual !!

y’ = +5.34 cm

Y’Y

p q

Virtual image

Converging mirror

FF CC

Wrench

Example 4.Example 4. How close must a girl’s face How close must a girl’s face be to a converging mirror of focal be to a converging mirror of focal length 25 cm, in order that she sees an length 25 cm, in order that she sees an erect image that is twice as large? (erect image that is twice as large? (M M = +2)= +2)

2 ; 2q

M q pp

AlsoAlso, ,

pfq

p f

2pf

pp f

ThusThus, ,

f = f = -2(p - f) = -2-2(p - f) = -2pp + + 22ff

f = f = -2-2pp + 2 + 2ff25 cm

2 2

fp p = 12.5

cm p = 12.5 cm

SummarySummary

y

Y’

R

q

p

f

2

Rf

2

Rf

1 1 1

p q f

1 1 1

p q f

The following equations apply equally The following equations apply equally well for both converging and diverging well for both converging and diverging mirrors.mirrors.

The following equations apply equally The following equations apply equally well for both converging and diverging well for both converging and diverging mirrors.mirrors.

Summary: Sign Summary: Sign ConventionConvention

1. Object distance p is positive for real objects and negative for virtual objects.

2. Image distance q is positive for real images and negative for virtual images.

3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.

4. The image size y’ and the magnification M of images is positive for erect images and negative for inverted images.

1 1 1

p q f

1 1 1

p q f

Summary: MagnificationSummary: MagnificationThe The magnificationmagnification MM of an image is the of an image is the ratio of the ratio of the image sizeimage size y’y’ to the to the object object sizesize yy..

Magnification:

'y qM

y p

yy and and y’ y’ are positive when erect; negative are positive when erect; negative inverted.inverted.

Obj.

Img.

Obj.

Img.

M = +2 M = -1/2

y y’

y y’

qq is positive when real; negative when virtual. is positive when real; negative when virtual.

MM is positive when image erect; negative is positive when image erect; negative inverted.inverted.

CONCLUSION: Chapter 34BCONCLUSION: Chapter 34BReflection and Mirrors II Reflection and Mirrors II

(Analytical)(Analytical)